How to make separate Lips control the mouth (Cavity) animation?
As described in this answer: How can I make two objects share the same vertex?, you can use Hooks to drive the deformation.

In Edit mode select a vertex on the mesh.

Press the Space Bar, type &quot;hook&quot; and select &quot;Hook to new object&quot;
This will create an empty that controls that specific vertex.

Exit edit mode.

Select the Newly created Empty, and while pressing Shift select the curve to make it the active object.

Enter edit mode and select a control point in the curve that matches the vertex you want to control.

Press the Space Bar and type &quot;vertex&quot;, select &quot;Make Vertex Parent&quot; (or Pres Ctrl+ P)
Now the control point in the curve moves the empty, and that in turn controls the vertex on the mesh.




Repeat the process for the rest of the vertices in the mesh so that you can control them with the curve.